A Classical Theory of Electromagnetism and Gravitation (I) Special Theory

By extending the Maxwell-Lorentz equations to five dimensions, it is shown that one is led to a simple unified theory of gravitational and electromagnetic phenomena. The generalized expressions for the force density and the work done per unit volume per unit time contain terms which correspond, respectively, to the effects of electric, magnetic, and gravitational fields. If it be assumed that no changes of physical quantities occur in the direction of the extra dimension so introduced, a special relativity theory of gravito-electromagnetic fields arises. Within this theory gravitational waves are propagated with the velocity of light, gravitational potential is invariant for Lorentz transformations, and gravitational force acts on the rest mass of a particle. The conservation laws of charge, momentum, and energy are shown to hold, but the last two yield a generalized Poynting vector, and a generalized expression for the energy density, both of which contain terms which depend on the gravitational field strengths. The finite velocity of propagation of gravitational waves leads at once to the result that an accelerated mass emits energy in the form of such waves. On the classical theory the radiation emitted by an electron has thus associated with it a small longitudinal gravitational component. Gravitational forces are shown to lead to a self-energy for an accelerated mass, and the classical radius of a mass $m$, corresponding to the classical radius of a charge, is $\frac{\mathrm{Gm}}{c^2}$, where $G$ is the gravitational constant.